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\begin{document}

\title{高等代数复习 }
\subtitle{A-重要定理 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年3月9日} }

\maketitle

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{冠名定理 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{enumerate}

\item  带余除法定理
\item  代数基本定理
\item  高斯引理
\item  艾森斯坦判别法
\item  高斯消元法
\item  克拉默公式
\item  行列式乘积公式
\item  若尔当标准形
\item  柯西-施瓦茨不等式
\item  斯密特正交化方法

\end{enumerate}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{标准形、标准型 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{enumerate}

\item  矩阵的行最简形
\item  矩阵的相抵标准形
\item  矩阵的相似标准形
\item  实数矩阵的合同标准形
\item  复数矩阵的合同标准形
\item  实数对称矩阵的正交相似标准形

%\vspace{0.3cm}

\item  实二次型的标准型
\item  复二次型的标准型
\item  实二次型的正交变换标准型

\end{enumerate}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A1. 带余除法定理 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理2.2.1. 在数域 $F$ 上的一元多项式环中，带余除法的结果是存在且唯一确定的。即设 $f(x),g(x)\in F[x]$, 则存在唯一的一对多项式 $q(x)$ 与 $r(x)$, 使得
$f(x)=g(x)q(x)+r(x)$, 且 $r(x)=0$ 或 $\deg r(x) < \deg g(x)$. 


\end{itemize}

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\begin{frame}{A2. 互素的充分必要条件 }

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\begin{itemize}

\item  定理2.3.3. 两个多项式 $f(x),g(x)$ 互素当且仅当存在两个多项式 $u(x),v(x)$ 使得
$f(x)u(x)+g(x)v(x) =1. $


\end{itemize}

\end{frame}

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\begin{frame}{A3. 存在重因式的充分必要条件 }

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%每页详细内容

\begin{itemize}

\item  定理2.5.2. 下述两个条件是等价的。
\begin{enumerate}
\item  多项式 $f(x)$ 没有重因式。
\item  多项式 $f(x)$ 与其导数多项式 $f\,'(x)$ 互素。
\end{enumerate}


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A4. 代数基本定理 }

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%每页详细内容

\begin{itemize}

\item  定理2.7.1. 任意次数大于等于1的多项式在复数范围内至少有1个根。

\vspace{0.5cm}

\item  定理2.7.4. 实数域上的不可约多项式如下：
\begin{enumerate}
\item   一次多项式 $a_0x+a_1,\quad a_0\neq 0$. 
\item   二次多项式 $a_0x^2+a_1x+a_2,\quad a_0\neq 0, a_1^2-4a_0a_2<0$. 
\end{enumerate}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A5. 高斯引理、艾森斯坦判别法 }

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%每页详细内容

\begin{itemize}

\item  引理2.8.1.(高斯引理) 两个本原多项式的乘积仍然是一个本原多项式。

\vspace{0.5cm}

\item  定理2.8.2.(艾森斯坦判别法) 设 $f(x)=a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ 是一个整系数多项式。设存在一个素数 $p$ 使得
$$p\nmid a_n, \quad p\mid a_{n-1},\cdots, p\mid a_1,\quad p\mid a_0,\quad p^2\nmid a_0, $$  
那么这个多项式在有理系数范围内是不可约的。

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A6. 行列式的两种计算 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理：行列式按行或按列展开计算，与按逆序数计算，结果是一样的。


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A7. 克拉默公式 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理：下述线性方程组
\begin{eqnarray*}
\left\{\begin{array}{rcl}
{\color{red}a_1}x_1+a_2x_2+a_3x_3+a_4x_4&=&{\color{blue}k},  \hspace{2cm} \times (A) \\
{\color{red}b_1}x_1+b_2x_2+b_3x_3+b_4x_4&=&{\color{blue}m},  \hspace{1.9cm} \times (B) \\
{\color{red}c_1}x_1+c_2x_2+c_3x_3+c_4x_4&=&{\color{blue}n},  \hspace{2cm} \times (C) \\
{\color{red}d_1}x_1+d_2x_2+d_3x_3+d_4x_4&=&{\color{blue}p},  \hspace{2cm} \times (D) 
\end{array}\right.
\end{eqnarray*}
的克拉默公式为
\begin{eqnarray*}
x_1= \frac{{\color{blue}k}A + {\color{blue}m}B + {\color{blue}n}C + {\color{blue}p}D}{{\color{red}a_1}A + {\color{red}b_1}B + {\color{red}c_1}C + {\color{red}d_1}D} 
= \frac{{\text{\small\lib 将系数行列式的第一列换成常数列}}} {{\text{\small\lib 系数行列式}}}. 
\end{eqnarray*}



\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A8. 高斯消元法 }

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%每页详细内容

\begin{itemize}

\item  定理：用行初等变换将增广矩阵化为行最简形，可得同解的线性方程组。


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A9. 线性方程组有解的充分必要条件 }

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%每页详细内容

\begin{itemize}

\item  定理：线性方程组 $AX=B$ 有解当且仅当 $R(A)=R(A,B)$, 即系数矩阵 $A$ 的秩等于增广矩阵 $\overline{A}=(A,B)$ 的秩。  

\vspace{0.5cm}

\item 定理：设线性方程组 $AX=B$ 有解，即有 $R(A)=R(A,B)$. 并设未知数个数是 $n$, 即列向量 $X$ 有 $n$ 个分量。则这个线性方程组的通解中有 $n-R(A)$ 个任意常数。

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A10. 初等变换与初等矩阵 }

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%每页详细内容

\begin{itemize}

\item  定理：
\begin{enumerate}
\item  对矩阵 $A$ 进行一次行初等变换，相当于在矩阵 $A$ 的左边乘以一个相应的初等矩阵。
\item  对矩阵 $A$ 进行一次列初等变换，相当于在矩阵 $A$ 的右边乘以一个相应的初等矩阵。
\end{enumerate}


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A11. 相抵标准形 }

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%每页详细内容

\begin{itemize}

\item   定理：设 $A$ 是一个 $m\times n$ 阶的矩阵，记 $r=R(A)$ 是矩阵 $A$ 的秩。则
\begin{enumerate}
\item   存在一系列的行初等变换和列初等变换，使得 
\begin{eqnarray*}
A \xrightarrow[\text{ }]{\text{一系列的初等变换 }}
\begin{pmatrix} E_r&O_{r,n-r} \\ O_{m-r,r}&O_{m-r,n-r} \end{pmatrix}. 
\end{eqnarray*}

\item  存在 $m$ 阶的初等矩阵 $P_1, \cdots, P_s$ 与 $n$ 阶的初等矩阵 $Q_1, \cdots, Q_t$ 使得 
\begin{eqnarray*}
P_s \cdots P_1A Q_1\cdots  Q_t 
=
\begin{pmatrix} E_r&O_{r,n-r} \\ O_{m-r,r}&O_{m-r,n-r} \end{pmatrix}. 
\end{eqnarray*}

\end{enumerate}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A12. 伴随矩阵与逆阵 }

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%每页详细内容

\begin{itemize}

\item  定理：若矩阵 $A$ 的行列式的值不等于零，即 $\det(A)\neq 0$, 则 $A$ 的逆阵为
\begin{eqnarray*}
A^{-1} = \frac{1}{\det(A)}A^*.
\end{eqnarray*}


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A13. 行列式乘积公式 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理：设 $A,B$ 是 $n$ 阶矩阵，则有 $\det(AB)=\det(A)\det(B)$. 


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A14. 线性方程组的解集 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

%\item  定理：设 $m\times n$ 矩阵 $A$ 的秩 $R(A)=r$, 则齐次线性方程组 $AX=0$ 的每个基础解系都正好有 $n-r$ 个向量。这也是解空间的维数。

%\vspace{0.5cm}

\item  定理：
设 $A$ 是一个 $m\times n$ 矩阵，设 $X=(x_1,\cdots,x_n)^t$ 是 $n$ 维列向量，由 $n$ 个未知数组成。
设向量 $\xi$ 是 $AX=\beta$ 的一个特解，即 $A\xi=\beta$. 
设向量组 $\{\eta_1,\cdots,\eta_t\}$ 是导出组 $AX=0$ 的一个基础解系。
则非齐次线性方程组 $AX=\beta$ 的解集为 
\begin{eqnarray*}
S &=& \{ X \in\mathbb{R}^n \mid AX=\beta \} \\  
    &=& \{ \xi + k_1\eta_1+\cdots + k_t\eta_t \mid k_1, \cdots, k_t\in\mathbb{R} \}. 
\end{eqnarray*}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A15. 线性代数与矩阵代数 }

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%每页详细内容

\begin{itemize}

\item  定理：数域 $F$ 上的 $n$ 维向量空间 $V$ 上的线性变换全体组成的代数，与数域 $F$ 上的 $n$ 阶矩阵全体组成的代数，在取定 $V$ 的一个基之后，是同构的。用数学符号表示，就是 $L(V)\cong M_n(F)$. 


\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A16. 线性变换可对角化的充分必要条件 }

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%每页详细内容

\begin{itemize}

\item  定理：设 $V$ 是数域 $F$ 上的一个 $n$ 维向量空间，
设 $f(\lambda)$ 是线性变换 $\sigma:V\to V$ 的特征多项式。
那么 $\sigma$ 在数域 $F$ 上是可对角化的充分必要条件是下述两个条件同时成立：

\begin{enumerate}
\item  多项式 $f(\lambda)$ 的根都在数域 $F$ 中。
\item  对多项式 $f(\lambda)$ 的每个根 $\lambda$, 本征子空间 $V_{\lambda}$ 的维数（几何重数）都等于 $\lambda$ 在这个多项式 $f(\lambda)$ 里的重数（代数重数）。
\end{enumerate}


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A17. 柯西-施瓦茨不等式 }

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%每页详细内容

\begin{itemize}

\item  定理：设 $V$ 是一个欧氏空间。设 $\alpha,\beta\in V$, 那么一定有
$$\langle \alpha,\beta \rangle^2 \le \langle \alpha,\alpha \rangle \cdot \langle \beta,\beta \rangle,$$
而且，当且仅当向量组 $\{\alpha,\beta\}$ 线性相关时，等号成立。



\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A18. 斯密特正交化方法 }

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%每页详细内容

\begin{itemize}

\item  定理：设有欧氏空间 $V$ 中的线性无关的一个向量组 $\{\alpha_1,\alpha_2,\cdots,\alpha_m\}$. 那么一定可以找到一个正交向量组 $\{\beta_1,\beta_2,\cdots,\beta_m\}$ 满足下述条件：
\begin{eqnarray*}
L(\alpha_1) &=& L(\beta_1), \\ 
L(\alpha_1,\alpha_2) &=& L(\beta_1,\beta_2), \\
\cdots && \cdots \\ 
L(\alpha_1,\alpha_2,\cdots,\alpha_m) &=& L(\beta_1,\beta_2,\cdots,\beta_m). 
\end{eqnarray*}


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A19. 正交变换的充分必要条件 }

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%每页详细内容

\begin{itemize}

\item  定理：欧氏空间到自身的线性变换是正交变换当且仅当这个线性变换在规范正交基下的矩阵是正交矩阵。

\end{itemize}

\end{frame}

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\begin{frame}{A20. 复数矩阵的合同标准形、复二次型的标准型 }

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%每页详细内容

\begin{itemize}

\item  定理：任意复数对称矩阵 $A$ 都合同于标准形 
\begin{eqnarray*}
P^tAP=\begin{pmatrix} E_r&O \\ O&O \end{pmatrix},
\end{eqnarray*}
其中 $r$ 是矩阵 $A$ 的秩。

\vspace{0.5cm}

\item  定理：设 $q_1(X)=X^tAX$ 是一个复二次型，则存在复数域上的非奇异的变量代换 $X=PY$ 使得
$$q_2(Y)=q_1(PY)=y_1^2+\cdots+y_r^2,$$
其中 $r$ 是二次型 $q_1(X)$ 的秩。


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A21. 实数矩阵的合同标准形、实二次型的标准型 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理：在实数范围内，任意实对称阵 $A$ 都合同于标准形
\begin{eqnarray*}
P^tAP=\begin{pmatrix} E_p&O&O \\ O&-E_q&O \\ O&O&O \end{pmatrix}.
\end{eqnarray*}

\vspace{0.5cm}

\item  定理：设 $q_1(X)=X^tAX$ 是一个实二次型，则存在实数域上的非奇异的变量代换 $X=PY$ 使得
$$q_2(Y)=q_1(PY)=y_1^2+\cdots+y_p^2-y_{p+1}^2-\cdots-y_{p+q}^2,$$
其中 $r=p+q$ 是二次型 $q_1(X)$ 的秩。


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A22. 正定矩阵的充分必要条件 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理：实二次型 $q(X)=X^tAX$ 是正定的当且仅当实对称矩阵 $A$ 的所有顺序主子式的值都大于零。


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{A23. 实对称矩阵正交相似于对角矩阵 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  定理：欧氏空间到自身的线性变换是对称变换当且仅当这个线性变换在规范正交基下的矩阵是对称矩阵。

\vspace{0.5cm}

\item  定理：设 $A$ 是实对称矩阵，则存在正交矩阵 $U$ 使得 $U^{\,t}AU$ 是对角阵。

\vspace{0.5cm}

\item  定理：实二次型 $q_1(X)=X^{\,t}AX$ 总能通过变量的正交变换 $X=UY$ 化成
$$q_2(Y) = q_1(UY) = \lambda_1y_1^2+\lambda_2y_2^2+\cdots+\lambda_n y_n^2,$$ 
其中系数 $\lambda_1, \lambda_2, \cdots, \lambda_n$ 是矩阵 $A$ 的全部特征值。




\end{itemize}

\end{frame}


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